Astronomy & Astrophysics manuscript no. schwdsphs_arxiv
April 11, 2013
c
ESO
2013
Model comparison of the dark matter profiles of Fornax,
Sculptor, Carina and Sextans
M. A. Breddels and A. Helmi
Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
arXiv:1304.2976v1 [astro-ph.CO] 10 Apr 2013
April 11, 2013
ABSTRACT
We use orbit based dynamical models to fit the 2nd and 4th moments of the line of sight velocity distributions of the
Fornax, Sculptor, Carina and Sextans dwarf spheroidal galaxies. Our goal is to compare dark matter profile models of
these four systems using Bayesian evidence. We consider NFW, Einasto and several cored profiles for their dark halos
and present the probability distribution functions of the model parameters. When considering each system separately,
we find there is no preference for one of these specific parametric density profiles. However, the combined evidence shows
that it is unlikely that all galaxies are embedded in the same type of cored profiles of the form ρDM ∝ 1/(1 + r 2 )β/2 ,
where β = 3, 4. For each galaxy, we also obtain an almost model independent, and therefore accurate, measurement
of the logarithmic slope of the dark matter density distribution at a radius ∼ r−3 , i.e. where the logarithmic slope of
the stellar density profile is −3. This slope ranges from ∼ −1.4 for Fornax to ∼ −1.1 for Sextans, both at ∼ 1 kpc.
All our best fit models essentially have the same mass distribution over a large range in radius (from just below r−3 to
the last measured data point). This remarkable finding likely implies much stronger constraints on the characteristics
that subhalos extracted from cosmological simulations should have in order to host the dSph galaxies around the Milky
Way.
Key words. galaxies: dwarf – galaxies: kinematics and dynamics
1. Introduction
According to galaxy formation theories dwarf spheroidal
galaxies are believed to inhabit massive dark matter halos.
Because of their large mass to light ratio these galaxies are
ideal to test fundamental predictions of the ΛCDM cosmological paradigm, since it is generally considered relatively
safe to neglect baryons in the construction of dynamical
models.
One of the strongest predictions from ΛCDM concerns
the dark matter density profile. Early simulations of dark
matter halos assembled in a cosmological context showed
that such a profile is accurately described by a two-sloped
form, now known as NFW profile (Navarro et al. 1996,
1997). More recently Einasto profiles have been shown
to provide a better fit (e.g. Navarro et al. 2010; Springel
et al. 2008), in particular for satellite galaxies (Vera-Ciro
et al. 2013). These predictions are made using dark matter only simulations and therefore neglect (by construction) the baryonic component. And although baryons are
sub-dominant in the total potential of the system (Walker
2012a), it has been suggested that they could play a role in
the evolution of dwarf spheroidal galaxies, for instance, in
modifying the internal orbital structure (Bryan et al. 2012)
and the overall density profile (Governato et al. 2012). The
complex evolution of baryons and its non-trivial interplay
with the host halo are difficult to model and not yet completely understood (see Pontzen & Governato 2012).
Another effect driving the internal dynamics of satellite halos is the tidal interaction with the main host. It
can change the density profile (Hayashi et al. 2003), the
geometrical shape of the mass distribution (Kuhlen et al.
2007), and also influence the kinematics of the embedded
stars (Łokas et al. 2010). Unfortunately these uncertainties
imply that even when the observations of the local dwarf
spheroidal galaxies are not consistent with being embedded in the halos predicted from pure dark matter N-body
simulations, this does not necessarily reflect a fundamental
problem of ΛCDM.
Thanks to their relative proximity, information for individual stars in the dwarf galaxies satellites of the Milky Way
are relatively easy to get. Sky positions are easily determined from photometry, and radial velocity measurements
are possible to estimate within an error of ∼ 2 km/s. Some
of the datasets compiled to the date include thousands of
individual members with line-of-sight velocities (Battaglia
et al. 2008, 2011, 2006; Helmi et al. 2006; Walker et al.
2009a) Proper motions of individual stars are currently still
too difficult to measure. Despite the fact that only three of
the total of six phase space coordinates are available from
measurements, it is possible to create dynamical models of
these systems that can be compared to these observables.
Following the method thoroughly described in Breddels et al. (2012) we set out to model Fornax, Sculptor,
Carina and Sextans with orbit-based dynamical methods
(Schwarzschild modeling) assuming they are embedded in
spherical halos. As extensively shown in the literature (e.g.
Cretton et al. 1999; Jardel & Gebhardt 2012; Richstone &
Tremaine 1984; Rix et al. 1997; Valluri et al. 2004; van den
Bosch et al. 2008; van der Marel et al. 1998) this method
allows to construct a non-parametric estimator of the distribution function. Among many, this method has one adArticle number, page 1 of 10
Name
Fornax
Sculptor
Carina
Sextans
NBatt
945(1)
1073(2)
811(3)
792(4)
NWalker
2633(5)
1541(5)
1982(5)
947(5)
Nmember
2936
1685
885
541
Table 1. Number of stars in the kinematic samples used in this
paper. Sources: (1) Battaglia et al. (2006), (2) Battaglia et al.
(2008), (3) Helmi et al. (2006); Koch et al. (2006); Starkenburg
et al. (2010), (4) Battaglia et al. (2011), (5) Walker et al. (2009a)
vantage over Jeans modeling, by not having to assume a
particular velocity anisotropy profile, therefore being more
general and thus less prone to biases associated to the assumptions. But even in this case there are other limitations
in the modeling such as the mass-anisotropy degeneracy. In
this work we use higher moments (fourth moment) of the
line of sight velocity distribution to get a better handle on
this degeneracy.
To compare how different shapes for the dark matter
profiles fit the data, we first need to establish a statistical framework. In this paper we do this in a Bayesian
way using the evidence (Mackay 2003). This method provides a natural way of comparing models in Bayesian inference and also makes it possible to combine the data of
all the dwarf spheroidals to test for example, if all dwarf
spheroidals could be embedded in a universal halo (Mateo
et al. 1993; Walker et al. 2009b). Furthermore, the shape
may give us hints to how the dwarf galaxy formed and the
anisotropy profile may be used to distinguish between evolutionary scenarios (see e.g. Helmi et al. 2012; Kazantzidis
et al. 2011; Mayer 2010).
This paper is organized as follows. We begin in §2
by presenting the data and all the ingredients needed to
do the model comparison. In §3 we present our dynamical and statistical methods. We present the results of our
Schwarzschild models for the four dSph in our sample in
§4.1, while the Bayesian model comparison is done in §4.2.
We discuss the implications of our results in §4.3 and conclude in §5.
2. Data
In this section we present the data that is used for fitting
our dynamical models. The radial velocity measurements
of the dwarf spheroidal galaxies come from Battaglia et al.
(2008, 2006); Helmi et al. (2006); Walker et al. (2009a) and
Battaglia et al. (2011). We plot radius versus heliocentric
velocity in Fig. 1 for each galaxy separately.
Figure 1 shows that each dSph suffers from foreground
(Milky Way) contamination. To remove this contamination and reliably identify member stars we have developed
a simple analytic model for the positional and kinematic
distribution of both foreground and the galaxy in question (along the lines of Battaglia et al. 2008, Breddels et
al. in prep). For each particular dataset1 , we assume that
the foreground has a constant surface density, and that the
dSph follows a specific stellar density profile. We also assume that the velocity distribution at each radius may be
1
For a given dSph there may be multiple datasets, and we treat
each independently because their sampling might be different.
Article number, page 2 of 10
modeled as sum of two Gaussians. The Gaussian describing the foreground has the same shape at all radii, while
that of the stars associated with the dwarf can have a varying dispersion with radius. Their relative amplitude also
changes as function of distance from the dwarf’s centre.
This model results in a determination of the relative contribution of member-to-non-member stars as a function of
velocity and radial distance R.
Based on this model we calculate the elliptical radius at
which the ratio of dSph:foreground is 3:1 (without using any
velocity information). We remove all stars outside of this
radius from the dataset. A particular star included in more
than one dataset is removed only when it it satisfies the
condition for all sets, for instance a star outside Re,cut, Batt ,
but inside Re,cut, Walker will not be discarded. This simple
clipping in elliptical radius cleans up part of the foreground
contamination. For completeness, the radii for all datasets
cleaned up in this way are presented in Table 2, as well as
the fit to the foreground model. The number of stars and
the sources are listed in Table 1.
For the resulting dataset, we compute the second and
fourth moment of the radial velocity as a function of circular radius as follows2 . We first define radial bins such
that each bin has at least 250 stars in the velocity range
vsys − 3σv , vsys + 3σv . If the last bin has less that 150 objects, the last two bins are merged. After this, we fit our
parametric model for the galaxy plus foreground for each
radial bin, to derive new velocity dispersions. Then for each
bin we do a 3σ clipping on the velocity using the new velocity dispersion, and from this selection we calculate the
second and fourth moments. The errors on the moments
are computed using Eqs. (17) and (19) in Breddels et al.
(2012). The second moment and the kurtosis3 are shown
in Fig. 2 for each galaxy, where the black dot corresponds
to the mean, and the error bars indicate the 1σ error bar.
The blue region shows the confidence interval for the NFW
model found in §4.1.
For the photometry we use analytic fits given by various
literature sources as listed in Table 3. Although the stellar
mass is sub-dominant in the potential, we do include its
contribution in the dynamic models and fix M/LV = 1, as
in Breddels et al. (2012).
3. Methods
3.1. Dynamical models
Our aim is to compare different models to establish what
type of dark matter profile best matches the kinematical
data of local dSph galaxies. Here we consider the following
profiles to describe the dark matter halos of the dwarfs in
2
Elliptical radii are only used for the clipping, for the rest of
the analysis we use the circular radius
3
The kurtosis is defined as γ2 = µ4 /µ22 , where µ4 is the fourth
and µ2 is the second moment of the line of sight velocity distribution.
M.A. Breddels & A. Helmi: Model comparison of dark matter profiles in local dSphs
Fig. 1. Radius versus line of sight velocity for Fornax, Sculptor Carina and Sextans. The horizontal lines show the borders of
the bins, the vertical lines denote the mean systemic velocity of the galaxy together with the ±3σ region.
Name
Fornax
Sculptor
Carina
Sextans
Re,max,Batt
(kpc)
1.82
1.37
0.86
1.86
Re,max,walker
(kpc)
2.21
1.65
0.96
1.65
µMW
(km/s)
41.1
17.9
70.9
67.5
σMW
(km/s)
38.9
47.4
62.5
74.5
µdwarf
(km/s)
55.1
110.6
222.9
224.3
σdwarf
(km/s)
12.1
10.1
6.6
7.9
Table 2. Parameters of the foreground plus dwarf galaxy model used for determining membership, as well as for deriving the
radial profiles for the second and fourth velocity moments for each dSph.
Name
Fornax
Sculptor
Carina
Sextans
distance
(kpc)
138(1)
79(3)
101(1)
86(1)
profile
Plummer2
Plummer3
Exponential4
Exponential4
scale radius
(kpc)
0.79
0.30
0.16
0.39
LV
×105 L⊙
100(2)
10(3)
2.4(4)
4.37(4)
Table 3. Distances, type of photometric profile used, scale radius and stellar luminosity used for the dynamic models. Sources:
(1) Mateo (1998), (2) Battaglia et al. (2006), (3) Battaglia et al. (2008), (4) Irwin & Hatzidimitriou (1995)
our sample:
ρ(r) =
ρ(r) =
ρ0
2,
x (1 + x)
ρ0
,
β/γ
(1 + xγ )
2 ′
ρ(r) = ρ0 exp − ′ xα − 1 ,
α
NFW
(1)
(cored) βγ-profile
(2)
Einasto
(3)
where x = r/rs and rs is the scale radius. Each model has
at least two unknown parameters rs and ρ0 . As we did in
Breddels et al. (2012), we transform these two parameters
to rs and M1kpc (the mass within 1 kpc). As discussed in
the Introduction, the NFW and Einasto models are known
to fit the halos dark matter distributions extracted from
cosmological N-body simulations. On the other hand, we
explore the βγ models to test the possibility of a core in the
dark halo. Note that, in comparison to the NFW profile,
the Einasto model has one extra parameter (α′ ), but here
we consider only two values for α′ = 0.2, 0.4 to cover the
range suggested by Vera-Ciro et al. (2013). On the other
hand, the βγ profiles have two extra parameters, but we
limit ourselves here to two different outer slopes (β = 3, 4)
and two different transition speeds between the inner and
the outer slopes (γ = 1, 2). Note that the βγ models have
a true core only for γ > 1, however in all cases the central
logarithmic slope vanishes, d log ρ/d log r = 0. However, we
loosely refer to these models as cored in what follows. Note
Article number, page 3 of 10
Fig. 2. Line of sight velocity moments for Fornax, Sculptor Carina and Sextans. For each galaxy we show the velocity dispersion
and the kurtosis. The black dots show the mean, and the error bars the 1σ error. The blue regions show the confidence interval
for the NFW fit, similar to Breddels et al. (2012).
that, with these choices, all of our profiles ultimately have
just two free parameters. The list of models explored and
their parameters are summarized in Table 4.
The orbit-based dynamical (Schwarzschild) models of
each dwarf galaxy are obtained as follows (see Breddels
et al. 2012, for a more detailed description). For each of
the dark halo profiles, with its own set of parameters, we
integrate a large number of orbits in the respective gravitational potential (including also the contribution of the
stars). We then find a linear combination of these orbits
that fits both the light and the kinematics. The orbital
weights found in this way have a physical meaning and can
Article number, page 4 of 10
be used to obtain the distribution function of the system.
As data we have the line of sight velocity moments (second and fourth depicted in Fig. 2), and the light profile
(Table 2). The best fit models (which give us the values
of the parameters for a specific dark matter halo profile)
are those that minimize the χ2 = χ2kin + χ2reg , under the
condition that the orbital weights are positive, and that
the observed light distribution
P is fit to better2 than 1% at
each radius. Here χ2kin = k (µ2,k − µmodel
) /var(µ2,k ) +
2,k
P
model 2
2
) /var(µ4,k ). The χreg is a regularization
k (µ4,k − µ4,k
term to make sure that the solution for the orbit weights
leads to a relatively smooth distribution function. Bred-
M.A. Breddels & A. Helmi: Model comparison of dark matter profiles in local dSphs
Name
NFW
core13
core14
core23
core24
einasto.2
einasto.4
Fixed parameters
β = 3, γ = 1
β = 4, γ = 1
β = 3, γ = 2
β = 4, γ = 2
α′ = 0.2
α′ = 0.4
Free parameters
M1kpc , rs
M1kpc , rs
M1kpc , rs
M1kpc , rs
M1kpc , rs
M1kpc , rs
M1kpc , rs
(7)
Table 4. Model names and their characteristic parameters of
the various dark matter density profiles explored.
dels et al. (2012) calibrated the amplitude of this term
for Sculptor. To have the regularization term for the
other dwarfs of the same relative strength, we note that
χ2reg ∝ 1/N , where N is the number of members with radial
velocities, since the χ2kin term also scales as 1/N . Therefore, normalizing its amplitude to that of Sculptor we get
χ2reg, dwarf = χ2reg, Scl × NScl/Ndwarf .
3.2. Bayesian model comparison
Background on Bayesian model comparison may be found
in Mackay (2003). For completeness we discuss it here
briefly, but we assume the reader is familiar with the basics
of Bayesian inference.
Given the data D and assuming a model Mi , the posterior for the parameters θi of this model is:
p(θi |D, Mi ) =
p(D|θi , Mi )p(θi |Mi )
.
p(D|Mi )
(4)
The normalization constant p(D|Mi ), also called the evidence, is of little interest in parameter inference, but is
useful in Bayesian model comparison. To assess the probability of a particular model given the data, we find
p(Mi |D) =
p(D|Mi )p(Mi )
,
p(D)
(5)
where we see the evidence is needed. In this case p(D) is
the uninteresting normalization constant, as it cancels out
if we compare two models:
p(Mi |D)
p(Mi )
p(D|Mi ) p(Mi )
=
= Bi,j
,
p(Mj |D)
p(D|Mj ) p(Mj )
p(Mj )
(6)
where Bi,j is called the Bayes factor. If we take the priors
on the different models to be equal (i.e. p(Mi ) = p(Mj )),
the ratio of the evidence (the Bayes factor Bi,j ) gives the
odds ratio of the two models given the data D.
Using these results we can perform model comparison between dark matter density profiles, i.e. M =
{Mnfw , MEinasto , ...}, and calculate for instance the odds
that a given galaxy is embedded in an NFW profile compared to an Einasto model, BNFW,Einasto .
Not only can we do model comparison on a single object,
but we may also test if our objects share a particular model
(e.g. they are all embedded in NFW halos). If our dataset
D consists of the observations of two galaxies, i.e. D =
D1 ∪ D2 and assuming the datasets are uncorrelated and
independent, we obtain:
p(Mi |D)
p(Mi )
p(D1 |Mi )p(D2 |Mi ) p(Mi )
=
= Bi,j,1 Bi,j,2
p(Mj |D)
p(D1 |Mj )p(D2 |Mj ) p(Mj )
p(Mj )
where each factor p(Dk |Mi ) should be marginalized over its
(own) characteristic parameters. From Eq. (7) we can see
that the odds ratio of the models and Bayes factor from
different measurements can be multiplied to give combined
evidence for a particular model.
Behind each p(Mi |D) is a set of orbit based dynamical
(Schwarzschild) models, obtained as described above. For
each of the models we calculate the evidence. Later on we
compare each model’s evidence to that of an NFW profile,
i.e. we compute the Bayes factor Bi,NFW , where i can be e.g.
Einasto. By definition BNFW,NFW = 1, and again assuming
equal priors on the different models, the Bayes factor equals
the odds ratio of the models, such that for Bi,NFW > 0,
model i is favored over an NFW profile.
4. Results
4.1. Schwarzschild models
As a result of our Schwarzschild modeling technique, we
obtain a two dimensional probability density function (pdf)
of the two parameters, M1kpc and rs , for each galaxy and
for each dark matter halo profile. In Fig. 3 we plot the
pdf for the cored models and the NFW and Einasto models
separately for each galaxy. The colored dots correspond
to the maximum likelihood for each of the corresponding
models as indicated by the legend. The contours show the
1 and 2σ equivalent confidence intervals (the 3σ contour is
not shown for clarity). For both Fornax and Sculptor the
parameters for all profiles are relatively well determined,
while for Carina and Sextans this is less so. This can be
attributed to the difference in sample size (and hence to
the smaller number of members) in these systems, which
has translated into fewer bins where the moments can be
computed (see Fig. 2). In general for all four galaxies the
scale radius for the cored profiles is found to be smaller
than that for the NFW/Einasto profiles. We come back to
this point in section 4.3.
Our model’s masses at r1/2 , the 3d radius enclosing half
of the stellar mass, are compatible with those of Wolf et al.
(2010). However, our results for Fornax do not agree with
those of Jardel & Gebhardt (2012). These authors prefer a
cored profile with a much larger scale radius, and their enclosed mass is smaller in comparison to Wolf et al. (2010).
We note that this might be partly related to the fact that
the amplitude of their line of sight velocity dispersion profile (see their Fig. 2) is slightly lower than what we have
determined here.
In Fig. 4 we overlay on the kinematic observables the
predictions from the best fit Schwarzschild models. We note
that all models provide very similar and virtually indistinguishable fits, especially for the 2nd moment. Some slight
differences are apparent in the kurtosis, but in all cases, the
differences are smaller than the error bars on the moments.
In general we find all anisotropy profiles to be roughly
constant with radius and slightly tangentially biased on average. We do not find significant differences between the
profiles for cored and NFW models (the reason for this will
become clear in §4.3). Fornax’ s anisotropy β ∼ −0.2 ± 0.2,
while Sculptor and Carina have on average β ∼ −0.5 ± 0.3.
For Sextans the anisotropy cannot be determined reliably,
β ∼ −0.3 ± 0.5. These values are compatible with those of
Article number, page 5 of 10
Sculptor
Carina
Sextans
cored
NFW/Einasto
Fornax
Fig. 3. Pdf for the two free parameters characterizing the dark halo profiles for each dSph galaxy obtained using Schwarzschild
modeling. The top row shows the pdfs with NFW/Einasto models, the bottom panel those for all cored models explored. The
contours show the 1 and 2σ confidence levels (the 3σ contour is not shown to avoid crowding the image).
Walker et al. (2007), which were derived using the spherical Jeans equation assuming a constant velocity anisotropy
profile.
4.2. Bayesian comparison of the models
We compute the evidence relative to the NFW using Eq. (6)
by integrating over the parameters (in our case the scale
radius and the mass) the pdfs shown in Fig. 3. We do
this for each dwarf galaxy and for all the models listed in
Table 4. The different Bayes factors are shown in Fig. 5.
Each set of bars shows the Bayes factors for the given dSph
galaxy (Bi,N F W,k ), whileQthe last set shows the combined
result (Bi,N F W,comb = k Bi,N F W,k ). We note that an
odds ratio between 1:2 till 1:3 is considered “Barely worth
mentioning” (Jeffreys 1998), and only odds ratios above
1:10 are considered “strong” evidence.
For each galaxy there is hardly any evidence for or
against an Einasto profile (with α′ = 0.2, 0.4) compared
to NFW. This is not unexpected since these profiles are
quite similar over a large region (Vera-Ciro et al. 2013).
Also in the case of the combined evidence the NFW and
Einasto are hard to distinguish. Comparing the NFW or
Einasto profiles for individual galaxies to the cored models, one cannot strongly rule out a particular model. For
Fornax, Sculptor and Carina, the γ = 2 models (where the
transition speed is fast) appear to be less likely, but this
is not the case for Sextans. However, when we look at the
combined evidence, i.e. we explore whether all dwarfs are
embedded in the same halos, such γ = 2 models are clearly
disfavored.
The results for Sculptor may be compared to those of
Breddels et al. (2012). In that paper, the authors found
that the maximum likelihood value for the central slope of
the density profile corresponded to a cored model. Since the
Article number, page 6 of 10
evidence is the integral of the pdf, and not directly related
to the maximum likelihood (except for a Gaussian distribution), we should not be surprised to find a slightly stronger
evidence for the NFW case here. In any case, the differences between the models are minor as shown graphically
in Fig. 4, and the evidence and the maximum likelihood
(marginally) favoring different models can be attributed
simply to not being able to distinguish amongst these.
4.3. A robust slope measurement
We now inspect in more detail the shape of the mass distributions found for the various best fitting models. We
are interested in exploring why the differences between the
various models as small as apparent in Fig. 4.
The top row of Fig. 6 shows the enclosed dark matter
mass for the best fit models (indicated by the solid dots in
Fig. 3) for each galaxy separately. We use the same color
coding as in Fig. 3, and also include the stellar mass in
black. The red-dashed vertical lines denote r−3 , the radius
at which the light density profile has a logarithmic slope
of −3, while the black line indicates r1/2 . This remarkable
figure shows that for each galaxy there is a region where the
mass distributions are truly almost indistinguishable from
one another. The different profiles, each characterized by
its own functional form, scale radius rs and mass M1kpc ,
conspire to produce a unique mass distribution. This region extends from slightly below r−3 to approximately the
location of the outermost data point (see bottom panel).
Here M (r) ∝ rx , where x ranges from 1.65 for Fnx, to 1.9
for Sextans.
In the middle row of Fig. 6 we plot the logarithmic slope
of the dark halo density distribution, where the black line
denotes the stellar density. Near the position where the
logslope of the stellar density is −3, all the best fit dark
M.A. Breddels & A. Helmi: Model comparison of dark matter profiles in local dSphs
Fig. 4. Similar to Fig. 2, except now we show the different best fit models for the various density profiles explored, which are
indicated with different colors (the color scheme is the same as in Fig. 3).
matter density profiles seem to reach a similar logslope, although the value of the slope varies from galaxy to galaxy.
The radius where the logslopes coincide lies, as expected,
inside the region where the mass distribution is well determined, since both quantities are related through derivatives.
To illustrate the distribution of the kinematic sample
with respect to the light, we plot in the bottom row of Fig.
6 the cumulative 2d radial distribution of the kinematic
data in black. The cumulative 2d radial distribution for
the light is plotted as the red histogram. All kinematic
datasets are more concentrated than the light, but no clear
trend is visible between the distribution of the kinematic
sample with respect to the light, and the exact location
where the logslope of most accurately determined.
The existence of a finite region where the mass is more
accurately determined has also been observed in the literature in works using MCMC in combination with Jeans
modeling. For example, it is visible in e.g. the right panel
of Fig 1. in Wolf et al. (2010), Fig. 18 in Walker (2012b),
and Fig. 10 in Jardel et al. (2013) for Draco, in the case of
a non-parametric density distribution with Schwarzschild
models.
The analysis of Wolf et al. (2010) used the light weighted
average of the velocity dispersion to relate the radius at
which the logslope of the light is −3, or the half light raArticle number, page 7 of 10
Fig. 5. Evidences for all models listed in Table 4, relative to the NFW case. The last column shows the combined evidence for
all galaxies together, and shows that the core23 and core24 are strongly disfavored.
Fornax
Sculptor
Carina
Sextans
Fig. 6. Top row: Enclosed mass as a function of radius for the different dark matter density profiles, with the stellar component
in black. Middle row: Logarithmic density slope as a function or radius, where the black curve corresponds again to the stellar
component. The red dashed line indicates r−3 , the radius at which the light profile has a logarithmic slope of −3, while the black
line indicates r1/2 , the radius at which half of the stellar mass in enclosed (in 3d). Bottom row: Cumulative density distribution
of the (2d) radial distribution of the data (black), and the light (red) showing the kinematic data is sampled more concentrated
towards the center.
dius, to the point where the mass is accurately (being independent on the anisotropy) and precisely (showing the
least uncertainty) determined. Our findings go beyond this
result. They suggest that whatever dynamical model or
method is explored, there is a better set of parameters to
Article number, page 8 of 10
describe the mass distribution of dSph galaxies. Let r−3
be the radius at which the logslope of the (3d) light distribution is −3. Since the mass is accurately determined in
this region, a natural parameter would be M−3 = M (r−3 ).
And since also the logslope at this radius is accurately de-
M.A. Breddels & A. Helmi: Model comparison of dark matter profiles in local dSphs
Fornax
Sculptor
Carina
Sextans
Fig. 7. Similar to Fig. 3, except now using M−3 and r−3 as parameters. Note that the contours for the NFW cannot go beyond
κ(r−3 ) ≥ −1.
ρ
termined, the next parameter should be κ−3 = dd log
log r |r=r−3 .
For any general model, if the values of β and γ are fixed,
this effectively makes rs a function of κ−3 .
Fig. 7 shows the pdf for the M−3 and κ−3 parameters for
both the NFW and core13 models for each galaxy, assuming a flat prior on these parameters in the domain shown
in this figure, except for the NFW profile which we limit
to κ−3 = −1.05, since for κ−3 ≥ −1 the scale radius is
unphysical. As can be seen from the pdf while there is still
uncertainty associated to the logslope at this radius, the
value is nearly model independent and therefore we believe
this value to be accurate, especially for Fornax and Sculptor (κ−3 = −1.4 ± 0.15 and κ−3 = −1.3 ± 0.12 respectively
for the NFW model). Note also that some uncertainties
might arise because the kinematics are not sampled exactly
according to the light.
These results also help us understand why we found that
the scale radii of the best fitting NFW profiles always to be
larger than those of the cored models (see Fig. 3). For the
NFW, we have
κ(r) =
d log ρ(r)
−2r
− 1,
=
d log r
r + rs
(8)
which can be easily solved for rs :
rs,nfw = −r
κ+3
.
κ+1
(9)
A similar solution can be found for the other parametric
models, for instance the γβ model gives:
rs,γβ = r
−κ
β+κ
−1/γ
.
(10)
If we now require that the slopes are the same at r−3 for
the NFW and core13 models, we find
κ
rs,nfw
=
,
rs,core13
κ+1
(11)
which is > 1 for κ < −1, explaining why the cored profiles
have smaller scale radii than the NFW profile, i.e. to get
the same logslope at the same location for the cored models,
their scale radius needs to be smaller than that of the NFW
profile. A similar result holds for the other cored models.
5. Conclusions
In this paper we have presented a comparison of dynamical models using different dark matter profiles for four
dwarf spheroidal galaxies in the Local Group, namely Fornax, Sculptor, Carina and Sextans. The model comparison
was done using Bayesian evidence. We have found that no
particular model is significantly preferred, and that all four
dwarf spheroidals are compatible with either NFW/Einasto
or any of the explored cored profiles. Only Sextans shows a
slight preference for cored models, but not with high odds.
Nonetheless, we find that it is very unlikely that all four
dwarf spheroidals are each embedded in a cored dark matter halo of the form ρDM ∝ 1/(1 + r2 )β/2 , with β = 3, 4.
Our best fit models however, conspire to produce the
same mass distribution over a relatively large range in radii,
from r−3 up to the last measured data point (which is often close to the nominal tidal radius obtained from fitting
the light profile). This M (r) ∼ rx , with x = 1.65 − 1.9,
is similar to that suggested by Walker et al. (2009b) albeit with a slightly steeper exponent. Another (related)
quantity that is robustly determined and independent of
the assumed dark matter density profile, is the logslope of
the density distribution at r−3 . We find for the dwarfs
in our sample, that this slope ranges from κ−3 ∼ −1.4 at
r−3 = 0.96 kpc for Fornax, to κ−3 ∼ −1.1 at r−3 = 0.98
kpc for Sextans.
These findings can be seen as an extension of the results
of Wolf et al. (2010), who showed that the mass at r−3
can be determined very accurately in a model independent
fashion. These authors demonstrated that this result might
be understood from the Jeans equation. Although we do
not have yet a solid mathematical explanation for our new
findings, we suspect that this might be obtained using the
virial theorem, which is effectively another, yet independent
moment of the collisionless Boltzmann equation.
In the near future, we will apply Schwarzschild modeling
to the same data but instead of the moments, we will use the
discrete individual measurements directly. This approach
should allows us to get the most out of the data, since no
information is lost. When binning, one loses spatial resolution, but also the higher moments of the line-of-sight velocity distribution are not included in the fitting procedure
because of their large and asymmetric errors. Furthermore,
the use of the full line-of-sight velocity distribution should
improve the precision of the anisotropy profile, which may
be an interesting quantity to discriminate formation scenarios.
Article number, page 9 of 10
This moments-to-discrete modeling step must be carried
out before deciding if and how much more data is needed
to discriminate among various dark matter density profiles.
Nonetheless, we have learned here that the functional form
of the mass distribution may be determined over a large
distance range, even when only a few hundred velocity measurements are available (as in the case of Sextans). However, the uncertainty on the value of the exact slope of the
density profile at e.g. r−3 is driven by the sample size.
An obvious next step is to establish if the subhalos extracted from cosmological simulations have the right characteristics to host the dSph of the Milky Way, now that
not only the mass, but also its functional form (1st and
2nd derivatives), of their dark halos have been determined
reliably.
Acknowledgments
We are grateful to Giuseppina Battaglia, Glenn van de
Ven and Remco van den Bosch for discussions that led to
the work presented here. We acknowledge financial support from NOVA (the Netherlands Research School for Astronomy), and European Research Council under ERC-StG
grant GALACTICA-24027.
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